Journal of Modern Physics, 2012, 3, 1918-1944
http://dx.doi.org/10.4236/jmp.2012.312242 Published Online December 2012 (http://www.SciRP.org/journal/jmp)
Imprints on CMB Angular Power Spectrum Modes from
Cosmological Reionization
Tiziana Trombetti1,2, Carlo Burigana1,3
1Istituto di Astrofisica Spaziale e Fisica Cosmica di Bologna, Bologna, Italy
2Dipartimento di Fisica, Università La Sapienza, Roma, Italy
3Dipartimento di Fisica e Scienze della Terra, Università degli Studi di Ferrara, Ferrara, Italy
Received September 11, 2012; revised October 15, 2012; accepted October 27, 2012
ABSTRACT
The accurate understanding of the ionization history of the Universe plays a fundamental role in modern cosmology. It
includes a phase of cosmological reionization after the standard recombination epoch, possibly associated to the early
stages of structure and star formation. While the simple “

-parametrization” of the reionization process and, in par-
ticular, of its imprints on the Cosmic Microwave Background (CMB) anisotropy likely represents a sufficiently accurate
modelling for the interpretation of current CMB data, a great attention has been recently posed on the accurate compu-
tation of the reionization signatures in the CMB for a large variety of astrophysical scenarios and physical processes.
The amplitude and shape of the B-mode Angular Power Spectrum (APS) depends, in particular, on the tensortoscalar
ratio, r, related to the energy scale of inflation, and on the reionization history, thus an accurate modeling of the reioni-
zation process will have implications for the precise determination of r or to set more precise constraints on it through
the joint analysis of E and B-mode polarization data available in the next future and from a mission of next generation.
In this work we review some classes of astrophysical and phenomenological reionization histories, beyond the simple

-parametrization, a present a careful characterization of the imprints introduced in all the CMB APS modes. We have
implemented a modified version of CAMB, the Cosmological Boltzmann code for computing the CMB anisotropy APS,
to introduce the predicted hydrogen and helium ionization fractions. We compared the results obtained for these models
for all the non-vanishing (in the assumed scenarios) modes of the CMB APS. Considering also the limitation from po-
tential residuals of astrophysical foregrounds, we discussed the capability of next data to disentangle between different
reionization scenarios in a wide range of tensor-to-scalar ratios.
Keywords: Cosmic Microwave Background Radiation: CMBR Polarization; Reionization; Gravitational Waves;
CMBR Polarization
1. Introduction
The accurate understanding of the ionization history of
the Universe plays a fundamental role in modern cos-
mology. The classical theory of hydrogen recombination
for pure baryonic cosmological models [1,2] has been
subsequently extended to non-baryonic dark matter mod-
els [3-5] and recently accurately updated to include also
helium recombination in the current cosmological sce-
nario (see e.g. [6] and references therein). Various mod-
els of the subsequent Universe ionization history have
been considered to take into account additional sources
of photon and energy production, possibly associated to
the early stages of structure and star formation, able to
significantly increase the free electron fraction, e
logical reionization phase may leave imprints in the
CMB providing a crucial “integrated” information on the
so-called dark and dawn ages, i.e. the epochs before or at
the beginning the formation of first cosmological structures.
For this reason, among the extraordinary results achieved
by the WMAP mission1, the contribution to the under-
standing of the cosmological reionization process has
received a great attention.
This work is aimed at a careful characterization of the
imprints introduced in the polarization anisotropy, with
particular attention to the B-modes. The amplitude of the
B-mode APS, whose detection represents a crucial in-
direct probe of the stochastic field of primordial gravity
waves (see e.g. [7,8]; see also [9-11]), crucially depends
on the ratio of the amplitudes of primordial tensor and
scalar perturbations
x
,
above the residual fraction after the standard
recombination epoch at rec. These photon and
energy production processes associated to the cosmo-

3
10  
3
10z

rTS, related to the energy scale

1http://lambda.gsfc.nasa.gov/product/map/current/.
C
opyright © 2012 SciRes. JMP

T. TROMBETTI, C. BURIGANA 1919
of inflation. On the other hand, the reionization history of
the Universe leaves imprints on the shape of the B-mode
as function of the multipole and on its amplitude.
Thus, an accurate modeling of the reionization process
will have implications for the precise determination of
or to set more precise constraints on it through the joint
analysis of all the CMB APS modes coming from data
available in the next future and from a mission of next
generation.

r
In Section 2 we briefly summarize the current obser-
vational information coming from WMAP on the cos-
mological reionization and describe its main imprints in
the CMB. In Sections 3 and 4 we review the main prop-
erties of the two classes of reionization models consid-
ered in this work, astrophysical and phenomenological,
respectively. For the latter kind of models, we explicitly
provide the relationship between model parameters and
the photon number and energy density injection. Section
5 concerns the numerical implementation we carried out
to include the considered reionization scenarios in our
Boltzmann code modified version. The experimental sen-
sitivity of on-going and future CMB anisotropy space
missions and the limitation coming from astrophysical
foregrounds are discussed in Section 6, while our main
results are presented in Section 7. Two Appendices (A
and B) report on some technical details of this work. A
list of acronyms widely used in this paper can be found in
Appendix C. Finally, in Section 8 we draw our conclusions.
2. Cosmological Reionization
To first approximation, the beginning of the reionization
process is identified by the Thomson optical depth,

.
The values of

compatible with WMAP 3 yr data, pos-
sibly complemented with external data, are typically in
the range (corresponding to a reionization
redshift in the range for a sudden reioniza-
tion history), the exact interval depending on the consid-
ered cosmological model and combination of data sets [12].
Subsequent WMAP data releases improved the measure
of
0.06 0.12
8.5 - 13.5
-

, achieving a 68% uncertainty of [13-
16]. Under various hypotheses (simple CDM model
with six parameters, inclusion of curvature and dark en-
ergy, of different kinds of isocurvature modes, of neu-
trino properties, of primordial helium mass fraction, or of
a reionization width) the best fit of
0.015


lies in the range
, while allowing for the presence of
primordial tensor perturbations or (and) of a running in
the power spectrum of primordial perturbations the best fit
of
0.086 - 0.089

lies in the range to . While
this simple “

09.0960.091 - 0.20

-parametrization” of the reionization
process and, in particular, of its imprints on the CMB
anisotropy likely represents a sufficiently accurate mod-
elling for the interpretation of current CMB data, a great
attention has been recently posed on the accurate com-
putation of the reionization signatures in the CMB for a
large variety of astrophysical scenarios and physical proc-
esses (see e.g. [17-25]) also in the view of WMAP ac-
cumulating data and of forthcoming and future experi-
ments beyond WMAP (see [26] for a review). In [27] a
detailed study of the impact of reionization, and the as-
sociated radiative feedback, on galaxy formation and of
the corresponding detectable signatures has been pre-
sented, focussing on a detailed comparison of two well
defined alternative prescriptions (suppression and filtering)
for the radiative feedback mechanism suppressing star
formation in small-mass halos, showing that they are
consistent with a wide set of existing observational data
but predict different 21 cm background signals accessible
to future observations. The corresponding signatures de-
tectable in the CMB have been then computed in [28].
Different scenarios have been investigated in [29] as-
suming that structure formation and/or extra sources of
energy injection in the cosmic plasma can induce a dou-
ble reionization epoch of the Universe at low (late
processes) or high (early processes) redshifts, providing
suitable analytical representations, called hereafter as
“phenomenological reionization histories”. In the late
models, hydrogen was typically considered firstly ion-
ized at a higher redshift (, mimicking a possible
effect by Pop III stars) and then at lower redshifts (z,
mimicking the effect by stars in galaxies), while in the
early reionization framework the authors hypothesized a
peak like reionization induced by energy injection in the
cosmic plasma at .
15z
6
2
some 10z
The sensitivity improvement of CMB polarization ex-
periments calls for a complete and accurate characteriza-
tion of all CMB APS, including the polarization B-modes,
from low to high multipoles. Among the various studies,
the reader could refer to the classical works by [30] and
[31] for the connection of B-modes with inflation and
their detectability, by [32] for the analysis of the kinetic
Sunyaev-Zel’dovich effect from mildly non-linear large-
scale structure, and by [33] for the computation of
B-modes induced by lensing.
In this work we review the relatively wide sets of as-
trophysical and phenomenological models introduced
above and present detailed computations of the signa-
tures they induce in the CMB temperature and polariza-
tion anisotropies. The methods described here can be also
used as guidelines for the implementation of any other
reionization scenario (see e.g. [34]). In particular, we
present precise computations of the polarization B-mode
APS for these models and compare them with the sensi-
tivity of on-going (Planck2, see [35]) and future (assum-
ing COrE3, see [36], as a reference) CMB space mis-
sions.
2www.rssd.esa.int/Planck.
3http://www.core-mission.org/.

T. TROMBETTI, C. BURIGANA 1920
Signatures in the CMB
The cosmological reionization leaves imprints on the
CMB depending on the (coupled) ionization and thermal
history. They can be divided in three categories4: 1) gen-
eration of CMB Comptonization and free-free spectral
distortions associated to the IGM electron temperature
increase during the reionization epoch [39-43], 2) sup-
pression of CMB temperature anisotropies at large mul-
tipoles, , due to photon diffusion, and 3) increasing of
the power of CMB polarization and temperature-polari-
zation cross-correlation anisotropy at various multipole
ranges, mainly depending on the reionization epoch, be-
cause of the delay of the effective last scattering surface.
The imprints on CMB anisotropies are mainly dependent
on the ionization history while CMB spectral distortions
strongly depend also on the thermal history. The reioni-
zation process mainly influences the polarization E and B
modes, and the cross-correlation temperature-polariza-
tion mode because of the linear polarization induced by
the Thomson scattering. The effect is typically particular
prominent at low multipoles , showing as a bump in
the power spectra, otherwise missing.



0.24 0.76


Through this note we assume a flat CDM model
compatible with WMAP, described by matter and cos-
mological constant (or dark energy) density parameters
and , reduced Hubble constant
m
0100 km/s/MpchH
t 0.7
0.73 , baryon density
20.022
bh , density contras84


index n
ure 5 K
, and adia-
batic scalar perturbations (without running) with spectral
0.95
s. We adopt a CMB background tem-
of 2.72 perat [44].
3. Astrophysical Reionization Models
The analysis of Ly

absorption in the spectra of the 19
highest redshift SDSS QSO shows a strong evolution of
the Gunn-Peterson Ly

opacity at [45,46]. The
downward revision of the electron scattering optical
depth to
6z
0.09 0.03



100MM
100
in the release of the 3 yr WMAP
data, confirmed by subsequent releases, is consistent with
“minimal reionization models” which do not require the
presence of very massive  Pop III stars
[47]. The above models can be then used to explore the
effects of reionization on galaxy formation, referred to as
“radiative feedback”.
A semi-analytic model to jointly study cosmic reioni-
zation and the thermal history of the IGM has been de-
veloped in [48]. According to Schneider and collabora-
tors, the semi-analytical model developed by Choudhury
& Ferrara, complemented by the additional physics in-
troduced in [49], involves: 1) the treatment of IGM in-
homogeneities by adopting the procedure of [50]; 2) the
modelling of the IGM treated as a multiphase medium,
following the thermal and ionization histories of neutral,
HII, and HeIII regions simultaneously in the presence of
ionizing photon sources represented by Pop III stars with
a standard Salpeter IMF extending in the range 1 -
e
Z
M
[51], Pop II stars with 0.2 Z
6z
 and Salpeter
IMF, and QSO (particularly relevant at ); 3) the
chemical feedback controlling the prolonged transition
from Pop III to Pop II stars in the merger-tree model by
Schneider; 4) assumptions on the escape fractions of
ionizing photons, considered to be independent of the
galaxy mass and redshift, but scaled to the amount of
produced ionizing photons. It then accounts for radiative
feedback inhibiting star formation in low-mass galaxies.
This semi-analytical model is determined by only four
free parameters: the star formation efficiencies of Pop II
and Pop III stars, a parameter, esc

, related to the escape
fraction of ionizing photons emitted by Pop II and Pop
III stars, and the normalization of the photon mean free
path, 0

, set to reproduce low-redshift observations of
Lyman-limit systems.
A variety of feedback mechanisms can suppress star
formation in mini-halos, i.e. halos with virial tempera-
tures vir , particularly if their clustering is taken
into account [52]. It is then possible to assume that stars
can form in halos down to a virial temperature of 104 K,
consistent with the interpretation of WMAP data [53]
(but see also [54]). Even galaxies with virial temperature
vir can be significantly affected by radiative
feedback during the reionization process, as the increase
in temperature of the cosmic gas can dramatically sup-
press their formation.
4
10 KT
4
10 KT
Based on cosmological simulations of reionization, [55]
developed an accurate characterization of the radiative
feedback on low-mass galaxies. This study shows that
the effect of photoionization is controlled by a single
mass scale in both the linear and non-linear regimes. The
gas fraction within dark matter halos at any given mo-
ment is fully specified by the current filtering mass,
which directly corresponds to the length scale over which
baryonic perturbations are smoothed in linear theory. The
results of this study provide a quantitative description of
radiative feedback, independently of whether this is
physically associated to photoevaporative flows or due to
accretion suppression.
Two specific alternative prescriptions for the radiative
feedback by these halos have been considered: 1) sup-
pression model—in photoionized regions halos can form
stars on ly if their circular velocity exceeds the critical
value
4Inhomogeneous reionization also produces CMB secondary anisot-
ropies that dominate over the primary CMB component for
and can be detected by upcoming experiments, like the ACT or ALMA
[37,38].
12
2B vkTm 
crit p




; here

is the mean mole-
cular weight,
p
4000lm is the proton mass, and T is the
average temperature of ionized regions, computed self-

T. TROMBETTI, C. BURIGANA 1921
consistently from the multiphase IGM model; 2) filtering
model—the average baryonic mass b
M
within halos in
photoionized regions is a fraction of the universal value
bbm
f , given by the fitting formula

3
13
C
12 1
bb
MM MM



f ,
where
M
is the total halo mass and C
M
is the total
mass of halos that on average retain 50% of their gas
mass. A good approximation for C
M
is given by the
linear-theory filtering mass,
  
12
1aaa




a
23 23
FJ
0
3d
a
MaaM

,
where is the cosmic scale factor,


32
2
Js
πMcG  4π3
is the Jeans mass,

is the average total mass density
of the Universe, and is the gas sound speed.
s
The model free parameters are constrained by a wide
range of observational data. Schneider and collaborators
reported the best fit choice of the above four parameters
for these two models that well accomplish a wide set of
astronomical observations, such as the redshift evolution
of Lyman-limit absorption systems, the Gunn-Peterson
and electron scattering optical depths, the cosmic star
formation history, and number counts of high redshift
sources in the NICMOS Hubble Ultra Deep Field.
c
4
10 K
v
6z
6z
15z
z
7
The two feedback prescriptions have a noticeable im-
pact on the overall reionization history and the relative
contribution of different ionizing sources. In fact, al-
though the two models predict similar global star forma-
tion histories dominated by Pop II stars, the Pop III star
formation rates have markedly different redshift evolu-
tion. Chemical feedback forces Pop III stars to live pref-
erentially in the smallest, quasi-unpolluted halos (virial
temperature ), which are those most affected by
radiative feedback. In the suppression model, where star
formation is totally suppressed below crit , Pop III stars
disappear at ; conversely, in the filtering model,
where halos suffer a gradual reduction of the available
gas mass, Pop III stars continue to form at , though
with a declining rate. Since the star formation and
photoionization rate at these redshifts are observationally
well constrained, the star formation efficiency and escape
fraction of Pop III stars need to be lower in the filtering
model in order to match the data. Therefore reionization
starts at in the filtering model and only 16% of
the volume is reionized at (while reionization
starts at in the suppression model and it is 85%
complete by ). For
10z
6z
20
10z

, QSOs, Pop II and
Pop III give a comparable contribution to the total
photo-ionization rate in the filtering model, whereas in
the suppression model reionization at is driven
primarily by QSOs, with a smaller contribution from Pop
II stars only.
7z
720z
15z
d
ee T
nct  

0.1017
The predicted free electron fraction and gas tempera-
ture evolution in the redshift range is very
different for the two feedback models. In particular, in
the filtering model the gas kinetic temperature is heated
above the CMB value only at .
The Thomson optical depth, , can be
directly computed for the assumed CDM cosmologi-
cal model parameters given the ionization histories:
06CF 0.0631 and 00G


 for the suppression
and the filtering model, respectively. Note that these
values are consistent with the Thomson optical depth
range derived from WMAP 3 yr (7 yr) data but with


12

 
2
difference among the two models, leaving
a chance of accurately probing them with forthcoming
CMB anisotropy experiments.
Fitting Astrophysical Histories
Since filtering and suppression models gives the redshift
evolution of ionization fraction and electron temperature
in a tabular way, as first step we derived these quantities
with appropriate analytical functions, by means of a spe-
cific fitting tool, Igor Pro (v. 6.21) [56]. In order to have
an accurate parametrization we divided the redshift in
bins such to minimize the

test given by the fit itself
(see Appendix A for details).
The results found using the complete set of functions
specific to each epoch of interest are plotted in Figure 1
for the ionization fraction and in Figure 2 for the elec-
tron temperature. Every graph shows the fitting functions
(dashed red line) and the corresponding reionization
model (solid black line).
The accuracy of the fit can be analyzed by means of
the percentile difference among theoretical data and fit-
ting functions, namely the ratios between the derived
functions and the models data, as shown in Figure 3.
From these plots one can see that the difference is always
<1%, a bit greater in the filtering model but, in any case,
the precision of the fit is always very high.
4. Phenomenological Reionization Models
We investigated two phenomenological double peaked
reionization models introduced by Naselsky & Chiang in
which the Universe was reionized twice at different ep-
ochs, the first one, late reionization model, at
by Pop III stars and then at by stars in large gal-
axies, and the second one, early reionization model, at
very high redshift (), induced for example by
decay of unstable particles, followed by reionization
echanisms at as predicted in the standard
10z
6z
100z
6z m

T. TROMBETTI, C. BURIGANA
JMP
1922
Figure 1. Ionization fraction: comparison between the tabulated data (solid black line) and the fit (dashed red line) for the
suppression (left panel) and filtering model (r ight pane l).
Figure 2. Electron temperature: comparison between the tabulated data (solid black line) and the fit (dashed red line) for the
suppression (left panel) and filtering model (r ight pane l).
Figure 3. Ionization fraction: ratios between the fit functions and the tabulated data for the ionization fraction (left panel)
and electron temperature (right panel) as function of redshift for the two different reionization histories.
picture. These models are based on the assumption that
extra ionizing sources can be described by the efficiency
characterizes the ionizing photon production rate:
of the ionizing photons production coefficient i

which
  
d=,
d
iib
nzn zHz
t
 (1)

T. TROMBETTI, C. BURIGANA 1923
where b is the baryon density and n

H
z

is the Hub-
ble parameter.
The late reionization history has been modeled as:


0
1
exp
iz






2
2
1.
re
m
re
zz
z
zzz







,
(2)
The two terms in the right hand of the equation de-
scribe the two ionizing epochs, with 01

 m
z

z

and free
parameters of the model, the width of the first
reionization epoch, the step function. The first peak
corresponds to a reionization fraction that decreases con-
siderably at re , while the second term characterizes
a reionization fraction which is monotonic increasing
with increasing time (or decreasing redshift).

z
For the early reionization history, the authors adopted
a representations in which the efficiency coefficient has a
Gaussian parametric form:

2
2;
re
zz
z






exp
iz
  (3)
again,

, re and are free parameters (clearly
different from those of the previous scenario).
zz
Suitable choices of these parameters allow us to model
different reionization scenarios, and with an accurate
balance of their values it is possible to obtain the desired
value of the Thomson optical depth. In principle, one can
derive the same value of

with different combinations
of these parameters, but the corresponding reionization
histories will be different. As result, the predictions for
the CMB APS will be also different. In order to present
results spanning a wider range of possibilities and also to
show the code versatility in ingesting parameter values
implying larger displacements in CMB APS predictions,
we exploited models not necessarily constrained by cur-
rent data. Assuming (in both astrophysical and phe-
nomenological models) the same cosmological parame-
ters defined in the previous section, we selected sets of
reionization parameters able to approximately reproduce
the optical depth found for the suppression model, an
astrophysical prescription (among those analyzed in this
work) in agreement with current constraints on

, in
order to focus on differences in APS even for models
characterized by the same Thomson optical depth5.
In particular, the late history was modeled by this set
of parameters:
3
0
6
1
06
9
1.3 10
10
5.3 100.1017.
10
11.95
0.025
re
LCF
re
z
m
zz








 






00
2.315
500 0.1017.
0.025
tot
reEE G
re
z
zz


 




(4)
Concerning the early scenario, since it is unable (alone)
to contribute to the actual optical depth having a substan-
tial high reionization redshift, we exploited its combina-
tion with the filtering model, such as the global optical
depth is exactly the same that figures out from the sup-
pression model, finding the following set of values:
(5)
In the case of early history, in order to satisfy the
above condition on z

(for the considered choice of

)
we investigated the dependence of the free variable

on the reionization redshift, re, in view of the broaden
range of values that this utmost parameter can explore in
comparison to a standard reionization model. As shown
in Figure 4, in spite of an initial linear dependence at
relatively low redshift, we observe a semi-parabolic be-
havior at higher redshifts.
z
The ionization fraction can be evaluated from the bal-
ance between the recombination and ionization proc-
esses:
 


2
d1
d
erecb eie
xTnxzx Hz
t
   (6)
where

0.6
1341 3
41010Ks cmT 


rec 
 is the re-
combination coefficient.
5Clearly, one could search for combinations of all relevant parameters
(i.e. related to the reionization model and to the underlying cosmologi-
cal model) that match current observational data, but this is not the
scope of the present work.
Figure 4. Dependence of the free parameter

on the
reionization redshift of the early history in order to have
the adopted value of

.

T. TROMBETTI, C. BURIGANA 1924
Assuming a curvature term k0


1m

 
, and so
the Hubble parameter is approximated by:

3
1.z 
0m
Hz H (7)
In Figure 5 we display the time evolution of the ioni-
zation fraction for all the models considered in this work.
In particular, the coupled early and filtering model is
plotted for three different cases of the reionization red-
shift. Note that, in order to have a constant optical depth,
we varied the parameter

assuming values

1.309 z2.315,  1.031,

500,350,200 when
re
respectively, giving a decreasing in the peak of the high
redshift region of e
x
, not linear with the reionization
redshift. The reason can be found analyzing the redshift
dependence of the recombination and ionization coeffi-
cients that are, respectively:
3,
rec b
Cnz

(8)
and

32
1,Hzz  
re
ze
ion ie
Cx   (9)
so that the higher is the lower is
x
(see Table 1
for details).
In these prescriptions the ionization fraction and elec-
tron temperature are provided in an analytical form and
connected through the dependence on the temperature of
the recombination coefficient rec

, that enters in the
time evolution equation of the ionization fraction. Since
Table 1. Redshift dependence of recombination and ioniza-
tion coefficients for decreasing reionization redshift and ad-
justed

model parameter when joining the early reioni-
zation history with the filtering model.

re z rec C ion C
2.315 500 8 1.25 10 4 2.59 10
1.031 350 7 4.29 10 3 6.75 10
3
3.70 10
1.309 200 6
8.00 10
Figure 5. Reionization fraction: comparison between mod-
els.
the relatively weak dependence, rec , the details
of the assumptions on the matter temperature are not par-
ticularly critical, although, in principle, for the active
phases out of equilibrium, they could play a non negligi-
ble role. For the active phase, we assume here, respec-
tively, a temperature profile mimicking the model by Cen
in the case of late processes and a temperature profile
given by the same Gaussian parametric form as in Equa-
tion (3) but with a peak temperature,
0.6
T 


p
T, as free pa-
rameter instead of

in the case of early processes, as
adopted by Burigana and collaborators in 2004. We use
here p. When the ionizing photons produc-
tion is negligible and no longer affects the ionization
history we assume the minimal ionization fraction usu-
ally derived in the absence of source terms, thus assuring
the continuity with the quiescent phases in the evolution
of the plasma properties.
4
610KT
Comoving Fractions of Injected Photon Number
and Energy Density
For many mechanisms of cosmological reionization, the
underlying physical process is usually characterized in
terms of an additional source of ionizing photons injected
in the plasma. We link here the parameters of the consid-
ered phenomenological histories to the comoving frac-
tions of injected photon number density and energy den-
sity. Defining the usual CMB photon number density

3
399 2.7KnT
if
zzz
00
, the comoving fractions of photon
number density injected in the redshift range
is given by:



4
0
d.
1
zib
i
zf
zn z
nz
nnz





1m zzz 

(10)
The second term, 1re

, appearing in
the late reionization history can be easily integrated:


10 10
1
0
0
0
d11,
1
zm
re bb
re
m
nn nzz
nnm
nz








52
1.12 10b nh


(11)
where 0b is the current baryon num-
ber density.
For the Gaussian term defining the early model and
appearing as first term in the late model, we find the fol-
lowing suitable approximation:


2
0
2
0
const exp d
1
zre
ib
zf
zz
n
nz
nnz z








 (12)
0
0
const π,
1
b
re
nz
nz


(13)


where “const” is the early parameter

or the late pa-
rameter 0

, respectively, and the assumption f
and i for the integration limits is made, a good
approximation to this aim for a peaked Gaussian shape.
z
z

T. TROMBETTI, C. BURIGANA 1925
The comoving fraction of injected photon energy den-
sity depends on the energy distribution function of ioniz-
ing photons. Given the comoving fraction of injected
photon number density and assuming the mean ionizing
photon energy, ion
E

, here assumed eV for nu-
merical estimates, one can compute of the comoving
fraction of injected photon energy density,
10

 . In the
case of the early history and for the Gaussian term of the
late history we have:

1
1
1 ;
7z





40
1.59 1010eV2.
ion
ET
n
n





 (14)
where for numerical estimates we can assume

early late
re . For the second term appearing
in the late reionization history, after a simple integration
we have:
11zz



11
1
1.601010
11 ,
m
re
E
z









10
0
eV 1
ion
b
n
m

 
00
aT  
y.
01DAJF
0.03%
.90ionizationf
(15)
where 4 is the current CMB photon energy
densit
In general, we used the routine of the NAG
Libraries for an accurate numerical cross-check of pre-
vious analytical formulas and approximations. For the
adopted parameters we found an agreement better than
.
5. Code Implementation
We modified CAMB [57], the cosmological Boltzmann
code (see also [58]) for computing the angular power
spectrum of the anisotropies of the CMB, in order to in-
troduce the ionization fractions evaluated according to
the astrophysical and phenomenological reionization mod-
els described in previous sections, alternative to the
reionization treatment originally implemented in CAMB.
Of course, the methods described here can be used as
guidelines for the implementation of any other astro-
physical reionization model (see Appendix B for further
details).
As a significant step forward with respect to previous
analyses, the emphasis of this work is posed to the exten-
sion to a first detailed characterization of the polarization
B-mode APS.
By implementing the source file ,
that defines the Reionization module, we are able to pa-
rametrize the desired reionization history and to supply
the corresponding ionization fraction as function of red-
shift [59].
re
Particular care must be taken to the normalization of
the quantities occurring in the history definition, such as
the ionization fraction. In CAMB the reionization frac-
tion is referred to the hydrogen, so, when allowing for
helium reionization, the global ionization fraction in the
case of complete ionization can be greater than one, fol-
lowing the relation:
full 1.
H
e
eH
n
n
 
full, 061.12721 CF
e
 
full, 001.12480 G
e
 
(16)
Instead, in the two astrophysical reionization models,
the global ionization fraction corresponding to the case
of complete ionization is for the
suppression model, and for the fil-
tering model.
To evaluate the total fraction in the case of the phe-
nomenological histories we have normalized it with
CAMB implementation, such as:
CAMB,1,
2
LE eH
ee
HH
n
f
xx
nf

 (17)
where
H
f
is the fraction of baryonic mass in hydrogen.
Assuming hydrogen abundance H, the global
ionization fraction in the case of complete ionization for
both histories is
0.76f
1.157895f,LE.

Furthermore, it is possible to fix in CAMB the Thom-
son optical depth parameter, and let the code estimates
the reionization redshift, or to choose the desired reioni-
zation redshift, and obtaining from dedicated function the

evaluation. While for the considered astrophysical
models

is known, we implemented a modified ver-
sion of this function to derive

according to the con-
sidered phenomenological model.
Astrophysical Models and Standard CAMB
In the framework of our adapted version of CAMB code
it can be useful to analyze an important differences be-
tween the various models we described. The crucial
characteristic resides in the interplay between different
physical processes at various epochs which contribute to
reionize the cosmological plasma. Every mixture of these
processes can lead to a distinctive scenario, as displayed
in Figure 5.
Since the standard CAMB traces a reionization that
follows the “single peak” evolution scheme, it can be
interesting to compare it with the suppression and filter-
ing models, investigating on their differences in the APS
and, in particular, in the sequence of acoustic peaks, at
various multipoles (see Figures 6 and 7).
To this aim we derived the temperature (TT), polariza-
tion (EE and BB), and cross-correlation (TE) APS of
CMB anisotropies for the two astrophysical reionization
histories, CF06 and G00, generically denoted as “mod-
els” in titles and the legends, and for the original version
of the code, denoted as “CAMB CF” or “CAMB G”,
assuming, respectively, an opt cal depth corresponding to i

T. TROMBETTI, C. BURIGANA 1926
Figure 6. Comparison between the models and CAMB in temperature (left panel) and E-mode polarization (right panel)
APS.
Figure 7. Comparison between the models and CAMB in B-mode polarization (left panel) and temperature-polarization
cross-correlation (r ight pane l) APS.
the value given by the theoretical model to which we are
comparing to. Note also that in the right panel of Figure
7 we display the module of the cross-correlation APS.
For simplicity, we neglect lensing in this case, and the
total  plotted represents the sum of scalar and tensor
contributions. The tensor to scalar ratio of primordial
perturbation, , is assumed here equal to .
C
r0.1
It is also very useful to analyse the relative difference
between results obtained with the astrophysical models
and the original CAMB, as shown in Figures 8 and 9,
defined by the relation:

Model CAMB
CAMB .
CC
CC




<

60 GHz 
Model
12 (18)
As before, note that in Figure 9 we display the module
of the difference of the cross-correlation APS.
The relative differences found in the case of the sup-
pression model are larger than in the case of the filtering
prescription, mainly in the polarization and cross-corre-
lation patterns of the APS. In addition, the difference is
remarkable at low multipoles, in particular at few
tens, i.e. at intermediate and large angular scales, as in-
tuitively expected since we are considering relatively late
reionization processes.
6. Sensitivity of CMB Measurements and
Foreground Emission
CMB experimental data are affected by uncertainties due
to instrumental noise (crucial at high multipoles, , i.e.
small angular scales), cosmic and sampling variance
(crucial at low , i.e. large angular scales) and from
systematic effects. Also, they are contaminated by a sig-
nificant level of foreground emission of both Galactic
and extragalactic origin. In polarization, the most critical
Galactic foregrounds are certainly synchrotron and ther-
mal dust emission, while free-free emission is negligible
in polarization and other components, like spinning dust
and haze, are still poorly known, particularly in polariza-
tion. Synchrotron emission is the dominant Galactic
foreground signal at low frequencies, up to ,
where dust emission starts to dominate. External galaxies

T. TROMBETTI, C. BURIGANA 1927
Figure 8. TT (left panel) and EE (right panel) APS: relative differences between the models and CAMB.
Figure 9. BB (left panel) and TE (right panel) APS: relative differences between the models and CAMB.
are critical only at high , and radiogalaxies are likely
the most crucial in polarization up frequencies ~200 GHz,
most suitable for CMB anisotropy experiments.

In this section we provide simple recipes aimed at
evaluating the levels of sensitivity of on-going and future
CMB space experiments and the kind of contamination
expected from foregrounds, and to make us able discuss
in the next section on the possibility to distinguish be-
tween different reionization scenarios in the framework
of current and future experiments, focussing in particular
on the pure polarization E and B-modes.
6.1. Sensitivity Measurements
The uncertainty on the angular power spectrum is given
by the combination of three components, CV, SV and
Instrumental Noise (N) [60]:

2
1.
CA
NCW




  
sky
sky
2
21 Cf




 (19)
Here
f
is the sky coverage,
A
is the surveyed
area,

is the instrumental rms noise per pixel, is
the total pixel number,  is the beam window function
that, in the case of a Gaussian symmetric beam, is:
N
W



2
exp1 ,
B
W  
 (20)
8ln2FWHM  
1f
being B the beamwidth which de-
fines the experiment angular resolution.
For sky

the first term in parenthesis defines the
cosmic variance, an intrinsic limit on the accuracy at
which the angular power spectrum of a certain cosmo-
logical model defined by a suitable set of parameters can
be measured with the CMB. It typically dominates the
uncertainty on the APS at low because of the small, 
21

m

W
, number of modes for each . The second
term in parenthesis characterizes the instrumental noise,
that never vanishes in the case of real experiments. Note
also the coupling between experiment sensitivity and
resolution, the former defining the low experimental
uncertainty, namely for  close to unit, the latter de-
termining the exponential loss in sensitivity at angular
scales comparable with the beamwidth.

T. TROMBETTI, C. BURIGANA 1928
In order to provide concrete estimates of these quanti-
ties, we consider Planck LFI and HFI channels at
, and COrE channels at
, i.e. at the frequen-
cies particularly suitable for CMB analyses because of
the combination of good experimental sensitivity and
resolution, and of relatively low foreground contamina-
tion. We adopt here the sensitivities and resolutions
summarized in the COrE white paper6.

70,100,1  
75,105,  

43,217GHz

95,225GHz

135,165,1
For each of the two projects we computed an overall
sensitivity value, weighted over the channels, defined by
22
,,
11
i
.
j
totj i


jPol

5%
(21)
where , and i states for the sensitivity of
each frequency channel, listed in Tables 2 and 3. FWHM
values of 13 and 14 are used to define the overall
resolution respectively of Planck and COrE in the com-
putation of the beam window function7.
T

Finally, to improve the signal to noise ratio in the APS
sensitivity, especially at high multipoles, we will apply a
multipole binning of in temperature APS, 15% in
TE cross-correlation and 30% in polarization APS, both
in E and B-modes.
Table 2. Instrumental sensitivity of Planck experiment .

GHz 

arcminFWHM


μKarcmin
T


μKarcmin
Pol

70 13 211.2 298.7
100 9.9 31.3 44.2
33.3
49.4
143 7.2 20.1
217 4.9 28.5
Table 3. Instrumental sensitivity of COrE experiment.

GHz 

arcminFWHM


μKarcmin
T

6.2. Parametrization of Residual Polarized
Foreground Contamination
The parametrization of the APS of Galactic thermal dust
and synchrotron emission adopted in this work is taken
from the results of WMAP 3-yrs [61] under the assump-
tion that these contributions are uncorrelated8, and is
expressed by:



μKarcmin
Pol

75 14 2.73 4.72
105 4.63
4.55
4.61
4.54
4.57
10 2.68
135 7.8 2.63
165 6.4 2.67
195 5.4 2.63
225 4.7 2.64
 

22
16565 ,
2π
sd
f
ore m
sd
C




  (22)
where
s
and stands for synchrotron and dust, and
the frequency
d

is expressed in GHz. The coefficients
characterizing the E and B-modes polarization APS are
slightly different, and are listed in Table 4.
In the next sections we will parametrize a potential re-
sidual from non perfect cleaning of CMB maps from
Galactic foregrounds simply assuming that a certain frac-
tion of the foreground signal at map level (or, equiva-
lently, its square at power spectrum level) contaminates
CMB maps. Of course, one can easily rescale the fol-
lowing results to any fraction of residual foreground
contamination. The frequency of 70 GHz, i.e. the Planck
channel where Galactic foreground is expected to be
minimum at least at angular scales above one degree,
will be adopted as reference.

0.1
10%
30%

2
μK
For what concerns extragalactic source fluctuations
[62,63], we will adopt the recent (conservative) estimate
of their Poissonian contribution to the APS [64] at 100
GHz9 assuming a detection threshold of Jy, to-
gether with a potential residual coming from an uncer-
tainty in the subtraction of this contribution computed
assuming a relative uncertainty of in the knowl-
edge of their degree of polarization and in the determina-
tion of the source detection threshold, implying a reduc-
tion to of the original level [65]. Except at very
high multipoles, their residual is likely significantly be-
low that coming from Galactic foregrounds.
7. Results
In this section we present the angular power spectra
Table 4. Parametrization of E and B mode polarization
power spectra of Galactic synchrotron and thermal dust
emission.
m

2
μK B-mode


E-mode m
sync 0.36 3.00.60.3 2.80.6
sync
dust 1.001.50.6 0.51.50.6

dust

6The nominal sensitivity of Planck is slightly better than that adopted
here thanks the slightly longer extension of the mission with both in-
struments, and the additional extension with the only LFI. Of course,
the real sensitivity of the whole mission will have to include also the
p
otential residuals of systematic effects.
7In fact, it is possible to smooth maps acquired at higher frequencies
with smaller beamwidths to the resolution corresponding to the lowest
frequency of each experiment.
8A more sophisticated treatment of the foreground power spectra takes
into account the correlation among the various foreground components.
9We adopt here a frequency slightly larger than that considered fo
r
Galactic foregrounds because at small angular scales, where point
sources are more critical, the minimum of foreground contamination is
likely shifted at higher frequencies.

T. TROMBETTI, C. BURIGANA
JMP
1929
2000
r
resulting from different prescriptions of the reionization
process and compare them with the sensitivities of
on-going and future space experiments including instru-
mental noise and fundamental statistical uncertainty.
Figures 10-13, show the temperature, polarization and
cross-correlation APS up to multipoles for a
range of values of the so-called tensor-to-scalar ratio, the
parameter (or
TS), which parametrizes the ratio
between the amplitudes of primordial tensor and scalar
perturbations. We also show the estimated cosmic and
sampling variance, and instrumental noise contributions
for the Planck (dotted lines) and COrE (dash-dotted lines)
experiments, assuming a sky coverage of 80% and a mul-
tipole binning of 5%, 30% and 15% depending on the
pattern.
As expected, the temperature APS does not exhibit a
remarkable dependence on the TS
 
parameter, as evi-
dent from the comparison between the panels in Figure
10. Comparing the considered reionization models, we
observe a difference between their relative power at high
and low multipoles: in particular, the early plus filtering
curves show more power at low and less at high .
Note that this double peaked reionization history has a
high redshift phase characterized by a remarkable ioniza-
tion fraction.
The E-mode power spectrum is shown in Figure 11.
Again, it is only weakly dependent on the T pa-
rameter, but for early plus filtering model that is widely
different from the other models up to the second acoustic
peak. Note also that, even if suppression and late (double
peaked) models have the same optical depth and cosmo-
logical parameters, their power spectra are significantly
different in both shape and power. For the late model, the
reionization bump is slightly shifted to higher multipoles
S
Figure 10. TT APS for the reionization histories: suppression, filtering, late double peaked, early plus filtering for different
values of tensor to scalar ratio (see plot legend and text for details).

T. TROMBETTI, C. BURIGANA 1930
Figure 11. EE APS for the reionization histories: suppression, filtering, late double peaked, early plus filtering for different
values of tensor to scalar ratio r (see plot legend and text for details).
with respect to the other histories.
The B-mode power spectrum (see Figure 12) plotted
here including also the lensing contribution, shows the
expected linear dependence on TS
r
70
1 
at low multipoles,
where the primordial signal dominates over the lensing
contribution, which, on the contrary, determines the
power at high almost independently of because of
the relative weight of primordial and lensing signal. The
tensor-to-scalar ratio parametrizes the B-mode anisot-
ropies from inflation resulting on a strong impact on the
shape of the observable acoustic peaks, in particular the
first one, affected by reionization and flattened for de-
creasing values of , and the second one, more related
to the recombination epoch and centered around .
These two peaks inevitably plays a fundamental role in
the study of the reionization history and consequently on
the first structures formation in the universe. Actually,
while tensors contribute to the E-mode polarization pat-
tern as well as scalars, the primordial B-modes are gen-
erated only by tensor perturbations, so detecting them
allow us to indirectly probe the stochastic field of pri-
mordial gravity waves (see e.g. [7,8]; see also [9-11]).
Meanwhile, to firmly achieve this aim is required a reso-
lution of at least [65], maybe lower than that of
Planck, but a significantly better sensitivity, as that fore-
seen for a next generation of experiments like COrE,
depending on the tensor-to-scalar ratio, while a resolu-
tion of few arcminutes would greatly help the disentangle
between primordial and lensing B-modes, being also
crucial for the study of other crucial topics in cosmology
(see e.g. [36]).
r
Typically, the reionization bump is stronger for the
suppression and the late (double peaked) models, weaker
for the filtering and the early plus filtering histories.

T. TROMBETTI, C. BURIGANA 1931
Figure 12. BB APS for the reionization histories: suppression, filtering, late double peaked, early plus filtering for different
values of tensor to scalar ratio r. Galactic synchrotron (dash-dotted blue line) and thermal dust (dash-triple-dotted cy an line)
polarized emission, and extragalactic point source fluctuations (solid orange line) and their potential residual (solid violet line)
as described in previous sec t ion (se e plot legend and text for details).
Comparing the panels in figure it is clear that the ideal
sensitivity of Planck is enough to detect the primordial
B-mode for tensor-to-scalar ratios above few 0.01

, in
particularly thanks to the information contained in the
reionization bump and up to the first acoustic peak. The
improvement foreseen for an experiment with a sensitiv-
ity like COrE could allow to reveal the primordial
B-mode polarization signal down to (or even
lower).
0.001r
The ultimate limitation comes from foregrounds. In
the case of the B-mode, we show an estimate of the con-
tamination by Galactic synchrotron and thermal dust po-
larized emission, and by extragalactic point source fluc-
tuations, parametrized as described in the previous sec-
tion. In all cases, the extragalactic signal is never domi-
nant except at very high multipoles, but still remaining
below the contribution by lensing, while Galactic fore-
ground may significantly contaminate the CMB measure,
especially at low and intermediate multipole.
The set of panels in Figure 13 presents the tempera-
ture-polarization cross-correlation APS, plotted in abso-
lute value. The plus sign at the top of each panel denote
positive correlation. Again, there is only a weak depend-
ence on the TS
400
parameter, and a substantial difference
between the considered evolutionary models, in particu-
ar for the early plus filtering history up to . l

T. TROMBETTI, C. BURIGANA 1932
Figure 13. TE APS for the reionization histories: suppression, filtering, late double peaked, early plus filtering (see plot leg-
end and text for details). Here is plotted the absolute value of the TE cross correlation, the “blue plus” indicates where the
are positive and each cusp is an inversion point in the sign of the themselves.
CC


Comparison between Models
In order to understand if different models can be distin-
guished, we analyze the relative differences between two
models and compare them with experimental sensitivity
and foreground residual estimates (properly normalized).
Since we are considering here several models, it is
useful to define a reference model to which divide their
differences as well as the experimental sensitivity and
foreground limitation. We assume here a CDM model
with the same cosmological parameters adopted in the
models under investigation, and an optical depth fixed by
the suppression history
R
ifCF . We compute its
power spectra using the standard CAMB.
 
1
8h Mpc

Note that, in principle, each model could be normal-
ized in order to match available CMB data in the desired
range of multipoles. Currently, temperature data play the
major role in this respect. The overall normalization of
the APS of a given model is related to the amplitude of
initial perturbations, or, equivalently, to the density con-
trast at a reference scale, typically assumed at ,
i.e. the parameter 8

, which is better determined when
CMB data are combined with galaxy surveys [66]. In
practice, this involves a multiplicative factor of the APS.
According to this choice, the relative differences between
two models could be more or less prominent at different
multipole ranges. In order to make our comparison be-
tween models dependent only on the APS shape and not
on the normalization adopted for each model, before of
computing their relative differences, we renormalize each
model to make its TT APS averaged over equal to 

T. TROMBETTI, C. BURIGANA 1933
that of the reference model.
We report in Figures 14-17 the relative differences
between models for the temperature anisotropies, polari-
zation EE and BB APS, and the temperature-polarization
cross-correlation, respectively. Each panel shows the
comparison between two models, normalized to the ref-
erence model defined above, for a wide set of the TS
100 - 300
70%10 - 15
parameter.
As anticipated, the tensor-to-scalar ratio does not af-
fect significantly the temperature anisotropies, and for
this reason in Figure 14 the curves appear almost su-
perimposed.
In addition, at the differences tend to be
approximately null, except for the comparison of the
early plus filtering model with all the others, because of
their dramatic difference at high redshifts in the ioniza-
tion fraction evolution.
Considering the whole multipole range, the largest dif-
ferences appear again in the comparison of the early plus
filtering model with all the other (later) histories, achiev-
ing a maximum level of at .
Note that we can discriminate only early processes
from other (later) models using only the CMB TT APS.
This can not be significantly improved with the future
Figure 14. Relative difference s between the astrophysic al and phenomenological reionization histories in temperature power
spectrum for all values of the r parameter assumed in this work (see plot legend and text for details).
R
i
f
C is the adopted
normalization power spectrum.

T. TROMBETTI, C. BURIGANA 1934
Figure 15. Relative differences between the astrophysical and phenomenological reionization histories in polarization
EE-mode power spectrum for all values of the r parameter assumed in this work.
R
i
f
C
r
r
200
5 - 25
100
is the adopted normalization power
spectrum. Potential residuals of Galactic fore gr ounds are also shown (see plot legend and te xt for details).
generation of experiments, the limitation coming essen-
tially from cosmic variance. Thus, polarization data are
crucial.
The comparison between the E-mode polarization
power spectra is more interesting (see Figure 15). As
evident, in particular in the three last panels in Figure 15
where the early plus filtering scenario is compared with
the others, the differences related to the parameter
emerge clearly. The greater is the greater are the rela-
tive differences. The large difference at is due
to the first early reionization phase, absent in the other
models.
In general, remarkable relative differences appear be-
tween all the models at  or even up to
when comparing the early plus filtering sce-
nario with the others. They are connected to the details of

T. TROMBETTI, C. BURIGANA 1935
Figure 16. Relative differe nces betw een the astr ophysic al and phenomenologic al reionization histories in polarization B-mode
power spectrum for all values of the r parameter assumed in this work.
R
i
f
C
SD0.03r
 is the adopted normalization power spectrum.
Potential residuals of Galactic foregrounds and extragalactic point source fluctuations are also shown (see plot legend and
text for details).
the ionization history at lower redshifts. We also plot a
potential residual contamination from Galactic fore-
grounds, in the plots. Note that a foreground
removal at a few per cent level of accuracy at map level
makes astrophysical contamination below the sensitivity
limitation of on-going and future space experiments. The
difference in nominal capability of Planck and COrE is
not so remarkable in this respect, although, in practice, a
significant improvement in sensitivity and frequency
coverage clearly will make much more robust and accu-
rate the foreground subtraction process.
More complex is the case of the B-mode polarization
power spectra, reported in Figure 16, where, for com-
pleteness, we display also an estimate of the potential
residual from extragalactic point source fluctuations
which is always negligible in comparison with the other
sources of limitation. For simplicity in these plots we
also choose a reference tensor to scalar ratio,

,
in order to represent the limit in the sensitivity of Planck
and COrE in respect to the relative signals of the histo-
ries, and to avoid an excessive overlapping of the curves.
The higher is its value, the bigger is the capability of the
instruments to discriminate between the models giving a
lower limiting curve. The same reference T para-
meter was applied for the contamination due to synchro-
tron and dust signals. As before, the early plus filtering
model largely differ from the others. Note that for the
B-mode the relative differences between the various
models remarkably depend on in all cases. The
Planck sensitivity strikingly depends on at low and
S
rr

T. TROMBETTI, C. BURIGANA 1936
Figure 17. Relative differences between the astrophysical and phenomenological reionization histories in temperature polari-
zation cross correlation power spectrum for all values of the r parameter assumed in this work (see plot legend and text for
details).
R
i
f
C

r
10
r
0.1r
is the adopted normalization power spectrum.
intermediate , because of the instrumental noise limi-
tation, while that of COrE is almost independent on ,
being essentially cosmic variance limited where the
lensing contribution does not dominate. The sensitivity
of COrE is such that we could distinguish between sup-
pression and filtering (or late) models in a certain range
of multipoles (around ) even for slightly lar-
ger than , while the early plus filtering history
can be distinguished from all the other histories on a
wider multipole region (or for even lower values of ).
The impact of residuals of Galactic polarized fore-
grounds clearly increases for decreasing tensor-to-scalar
ratio, as expected in the comparison of relative differ-
ences between models. Only for a foreground
subtraction at 3% accuracy level in the map is enough to
discriminate between each couple of considered models,
while the early plus filtering history can be distinguished
from all the other histories even with a less accurate
foreground subtraction, thanks to its prominent differ-
ences at intermediate multipoles.
r
3
10 
Finally, relative differences in the temperature-po-
larization cross-correlation APS are also very large on
very wide ranges of multipole when comparing the early
plus filtering history with all the other models (see Fig-

T. TROMBETTI, C. BURIGANA 1937
ure 17), filtering and late models show remarkable dif-
ferences (larger than ) at low multipoles, while the
suppression model differ from the late and filtering mod-
els only on a very small range of multipoles around
. The “spikes” appearing at high multipoles are
due to little shifts of the multipoles corresponding to the
change of sign of the cross-correlation spectra for the
different considered models. The difference in nominal
capability of Planck and COrE is not so remarkable for
the TE mode.
80%
10

8. Conclusions
The inclusion of astrophysically motivated ionization and
thermal histories in numerical codes is crucial for the
accurate prediction of the features induced in the CMB,
for constraining reionization models with CMB data, and
to exploit current and future high quality CMB data with
great versatility to accurately extract cosmological in-
formation.
We have implemented a modified version of CAMB,
the Cosmological Boltzmann code for computing the
APS of the anisotropies of the CMB, to introduce the
hydrogen and helium ionization fractions predicted in
two astrophysical reionization models, i.e. suppression
and filtering model, in two classes of phenomenological
reionization histories, involving late or early reionization,
and in their combination, as alternative to the original
implementation of reionization in the CAMB code, and
beyond the simple
9. Acknowledgements
We acknowledge the use of the Legacy Archive for Mi-
crowave Background Data Analysis (LAMBDA). Sup-
port for LAMBDA is provided by the NASA Office of
Space Science. We warmly thank Enrico Franceschi for
his technical advices in TeX compilation. We acknowl-
edge support by ASI through ASI/INAF Agreement
I/072/09/0 for the Planck LFI Activity of Phase E2 and
by MIUR through PRIN 2009 grant no. 2009XZ54H2.
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Appendix
A. Fitting Ionization and Thermal Histories
The considered astrophysical reionization models, as
well as others published in the literature, provide ioniza-
tion and thermal histories in tabulated form. The cosmo-
logical analysis of CMB anisotropy data largely relies on
Boltzmann codes for computing the angular power spec-
trum in temperature, polarization, and cross-correlation
modes under general conditions. The inclusion in such
codes of reionization histories beyond the simplistic
phenomenological approximations already implemented
in the publicly available codes, allows to achieve more
accurate predictions, of particular interest for the analysis
of polarization data. Having functional descriptions of
the evolution of the ionization fraction allows to speed-
up computation and improves code versatility with re-
spect to the use of interpolation of tabulated grids. Al-
though in this context only the ionization history is rele-
vant, we report here for completeness also the results
concerning the thermal history.
The software we used to fit to the suppression and fil-
tering model is Igor Pro (v. 6.21), an integrated program
for visualizing, analyzing, transforming and presenting
experimental data, such as curve-fitting, Fourier trans-
forms, smoothing, statistics, and other data analysis, im-
age display and processing, by a combination of graphi-
cal and command-line user interface. For completeness,
we report here all the functional forms we found particu-
larly suitable to represent the considered ionization and
thermal histories. Their compilation could be also useful
as guideline to fit other kinds of reionization history.
More details about the usage of this software to the cur-
rent aims and the complete list of parameters of the be-
low functions are given in [59].
A.1. Fitting Functions for the Reionization Histories
We report here below the redshift intervals and the cor-
responding analytic functions found in the case of the
suppression model.



0, 3.8—Polinomial Function of 6 order: z
3
ficien t s
.082246,
0.041472,
0.0012916,
6
0
01
2
45
5
6
with coef
1.12751, 0
0.083182,
0.010615,
5.5608 10 .
i
re i
i
z 








 






3.8,6 —Polinomial Function of 5 order: z
5
0
01
with
14.708,
i
re i
i
z 




coefficients
16.69,
23
45
6.8995, 1.4055,
0.14167, 0.0056647.


 
 



6,9z—Polinomial Function of 5 order, with
coefficients:
012
34 5
8.6358, 6.7374,1.821,
0.24108,0.015686, 0.00040015.

 
 
 



9, 1 2.4—Log-Normal Function: z

2
2
01
3
01
23
ln
expwithcoefficients
1.0061, 0.83887,
14.107, 0.26413.
re
z 
 





 








12.4,14.2 —Sigmoidal Function: z
1
0
2
3
01
23
with coefficients
1exp
1.1468, 1.153,
11.39,1.2238.
re z

 



 







14.2,16.8 —Hill Equation: z


2
22
010
3
01
23
withcoefficients
0.0021632, 0.46249,
13.862, 12.952.
re z
z


 


 
 
 



16.8,18 —Decaying Exponential Function (exp-
XOffset):
z
3
01
2
6
01
23
expwith coefficients
7.2284 10,0.010169,
1.054, 16.8.
re z

 





 






18, 20—Hill Equation, with coefficients: z
01
23
0.00048712, 0.013733,
24.474, 17.051.



 



20,20.2—Linear Function: z
01
01
with coefficients
0.0052133, 0.000223.
re z







20.2, 23—Decaying Exponential Function (exp-
XOffset), with coefficients:
z
01
23
0.00023877, 0.00047086,
0.85918, 20.2.







23,30—Log-Normal Function, with coeffi-
cients:
z

T. TROMBETTI, C. BURIGANA 1942
6
.6891 10,


01
23
0.00025689, 9
25.421, 0.10111.




We report here below the redshift intervals and the
corresponding analytic functions found in the case of the
filtering model.



0, 3.8—Polinomial Function of 9 order: z
5
ients
5,
489,
856,
045895,
1.4397 10.


9
0
01
23
45
67
89
withcoeffic
1.1245, 0.1049
0.19535,0.25
0.20598,0.098
0.028032, 0.0
0.00040036,
i
re i
i
z 








 
 
 
 




3.8,6—Log-Normal Function, with coefficients: z
0.7941,
1.7469.


01
23
1.8818,
6.3915,





6, 6.2—Linear Function, with coefficients: z
1
0.1981.
0
2.2768,

 



6.2,9—Hill Equation, with coefficients: z
01
23
2.7323,
5.6876.


0.1805,
3.6558,







9, 1 1.6—Sigmoidal Function, with coefficients: z
1.1441,
1.4126.


01
23
1.1438,
7.3904,





11.6,13 —Power Function: z
6
1
efficients
.052910 ,
2
01
0
2
withco
0.0089383, 2
7.0525.
re z





 




13,15—Power Function, with coefficients: z
2
9.5468.  
8
01
0.0015104,9.163710 ,   



15,17—Decaying Exponential Function (exp-
XOffset), with coefficients:
z
0.003812,
01
23
0.00013226,
1.1503,15.







17,19.6 —Decaying Exponential Function (exp-
XOffset), with coefficients:
z
1
.00057014,

0
23
0.00023759, 0
0.98278, 17.






19
5
1
.6107 10,


.6, 22.2—Decaying Exponential Function (exp-
XOffset), with coefficients:
z
0
23
0.00023514, 4
1.3061, 19.6.







22.2,30 —Decaying Exponential Function (exp-
XOffset), with coefficients:
z
01
23
0.052228, 0.051987,
26607, 22.2.




A.2. Fitting Functi ons fo r the Temperature Histories
We report here below the redshift intervals and the cor-
responding analytic functions found in the case of the
suppression model.



0,3 . 8—Polinomial Function of 8 order: z
8
0
01 2
345
67 8
withcoefficients
3870, 6793.7,7221.8,
15962, 13644,6419.7,
1734.7,252.65,15.379.
i
re i
i
TTz
TT T
TTT
TT T


 






3.8,5.8 —Sigmoidal Function: z
1
0
2
3
01
23
withcoefficients
1exp
20934, 8982.3,
4.0373, 0.45319.
re T
TT Tz
T
TT
TT
 








5.8,13—Hill Eqation: z

2
22
010
3
01
23
with coefficients
204.2,12105,
10.529,11.612.
T
re TT
z
TTTT
zT
TT
TT
 





13,16—Hill Equation, with coefficients: z
01
23
60.289, 8498.4,
12.484,12.23.
TT
TT
 




16,16.8 —Decaying Exponential Function (exp-
XOffset):
z
3
01
2
0123
expwith coefficients
0.78206, 229.94,1.0994,16.
re Tz
TTT T
TTTT


 




16.8,18.6 —Hill Equation, with coefficients: z
01
23
11.543,705.41,
19.532,15.338.
TT
TT





18.6, 20.2—Log-Normal Function: z


2
2
01
3
01 23
ln
expwith coefficients
16.488,34.915, 17.003,0.082807.
re
zT
TTT T
TTTT



 




 



20.2, 21—Sigmoidal Function, with coefficients: z
01
23
23.26, 8.5661,
19.785, 0.41168.
TT
TT





21, 22—Log-Normal Function, with coefficients: z

T. TROMBETTI, C. BURIGANA 1943
01
23
15.453,
21.501,
TT
TT


0.54634,
0.038693.




22,30—Log-Normal Function, with coefficients: z
46.863,
0.82554.


01
23
60.399,
18.997,
TT
TT
Finally, we report here below the redshift intervals and
the corresponding analytic functions found in the case of
the filtering model.



0, 3.8—Polinomial Function of 8 order, with
coefficients:
z
2
5
8
8613.8,
6902.3,
18.706.
T
T
T



01
34
67
3794.4, 6241,
17286, 14511,
1923.6, 292.58,
TT
TT
TT






3.8,6.2—Sigmoidal Function, with coefficients: z
23
9256.6,
0.50552.


01
21591,
3.9293,
TT
TT



6.2,10—Log-Normal Function, with coefficients: z
23
51883,
1.3666.


01
52588,
12.187,
TT
TT



10,11—Decaying Exponential Function (exp-
XOffset), with coefficients:
z
01
1989.4,
10.
T

23
210.57,
1.8033,
T
TT




11, 15—Sigmoidal Function, with coefficients: z
4674.2,
1.2229.


01
23
4677.4,
9.2859,
TT
TT



15,18—Sigmoidal Function, with coefficients: z
170.14,
0.84978.


01
23
180.89,
13.867,
TT
TT



18,21—Log-Normal Function, with coeffi-
cients:
z
2.5969,
0.11421.

01
23
14.387,
18.74,
TT
TT



21,30—Log-Normal Function, with coeffi-
cients:
z
01
23
37.272,
0.66903.


e
8.1645,
53.425,
TT
TT
B. Code Implementation in CAMB
We give here some details on the routines we imple-
mented in CAMB to make it able to reproduce the con-
sidered histories for the ionization fraction and electron
temperature, summarizing the main innovations included
in the modules of interest.
B.1. Subroutine Modifications in Reionization
Module
The first improvement concern the type Reionization
Params with the inclusion of a new string variable, history,
through which the user can discriminate between the
models, stored in the settings parameters file params.ini.
The main function of the original module, Reioni-
zation_xe, has been now written for each history, Reioni-
zation_xeCF, Reionization_xeG, Reionization_xeL, Reio-
nization_xeE. All of them retrieve the analytical ionization
fraction for each redshift bin of interest.
With this approach is not necessary to parametrize

in terms of the variable WindowVarMid:

32
1,yz (23)
where the exponent is set with the constant Reioni-
zation_zexp.
Some initial parameter values have been revisited too,
such as the maximum redshift at which e

varies, fixed
to instead of to take into account high red-
shift reionization phases, necessary for the early history,
and the corresponding initial scale factor astart, inver-
sely proportional to the redshift:
700 40
1.
1
az  (24)

In the function the mini-
mum number of time steps between
Re _ionizationtimesteps
_
ta and
, the relevant times for the reionization
process, has been incremented to 1000 , while for the
implementation of the adopted ionization histories the
functions listed below are no longer necessary:
u start
_tau complete
Re _ionization dopt

Re _,ionizationGetOptDepthReion R

Re _,ionizationzreFromOptDepthReion R

Re _,ionizationSetFromOptDepthReion R
0
0d,
eT
an



depth z, the subroutine that ex-
presses the integral optical depth in terms of the scale
factor,
 eionHist , the
routine which evaluates the integral of the optical
depth in the redshift interval

max
0, reion z,
 eionHist ,
a general routine to find the re
z parameter given
optical depth,
 eionHist ,
the subroutine that calculates the redshift of reioni-
zation.
This set of function, in fact, is related to the optical
depth definition, i.e.
  (25)

as computed in the standard CAMB, while for the histo-
ries under examination we have implemented specific
codes to evaluate it, both in our modified CAMB version
and as independent codes.
B.2. Subroutine Modifications in ThermoData
Module
The Thermodata module, implemented in modules.f90
source file, contains the subroutine inithermo(taumin,
taumax), which evaluates the unperturbed baryon tem-

T. TROMBETTI, C. BURIGANA 1944
e
perature and ionization fraction as function of time. If
there is reionization, the function discriminates between
the models, smoothly increases
CAMB: Code for the Anisotropy Microwave Back-
ground

to the requested
value and sets the to the value im-
posed by the corresponding model.
APS: Angular Power Spectrum
__opt depthactualCF06: Suppression Reionization Model
G00: Filtering Reionization Model
CV: Cosmic Variance
C. List of Acronyms SV: Sampling Variance
IR: InfraRed
CMB: Cosmic Microwave Background ACT: Atacama Cosmology Telescope
COrE: Cosmic Origins Explorer ALMA: Atacama Large Millimeter/Submillimeter Array
WMAP: Wilkinson Microwave Anisotropy Probe FWHM: Full Width Half Maximum
IGM: Inter Galactic Medium HFI: HFI Frequency Instrument
SDSS: Sloan Digital Sky Survey LFI: Low Frequency Instrument
QSO: Quasar CDM: Cold Dark Matter
IMF: Initial Mass Function